2
5
8
13
Correct answer is B
\((m_{1} v_{1} + m_{2} v_{2}) = (m_{1} + m_{2})v\) (Inelastic momentum)
\(8x + (5 \times 2) = (8 + 5) \times 3.85\)
\(8x + 10 = 13 \times 3.85 = 50.05\)
\(8x = 50.05 - 10 = 40.05\)
\(x \approxeq 5 m/s\)
Find the area of the circle whose equation is given as \(x^{2} + y^{2} - 4x + 8y + 11 = 0\)
\(3\pi\)
\(6\pi\)
\(9\pi\)
\(12\pi\)
Correct answer is C
Equation of a circle: \((x - a)^{2} + (y - b)^{2} = r^{2}\)
Given that \(x^{2} + y^{2} - 4x + 8y + 11 = 0\)
Expanding the equation of a circle, we have: \(x^{2} - 2ax + a^{2} + y^{2} - 2by + b^{2} = r^{2}\)
Comparing this expansion with the given equation, we have
\(2a = 4 \implies a = 2\)
\(-2b = 8 \implies b = -4\)
\(r^{2} - a^{2} - b^{2} = -11 \implies r^{2} = -11 + 2^{2} + 4^{2} =9\)
\(r = 3\)
\(Area = \pi r^{2} = \pi \times 3^{2}\)
= \(9\pi\)
97.9°
79.7°
63.4°
36.4°
Correct answer is B
\(m . n = |m||n|\cos \theta\)
\((3i + 4j) . (2i - j) = 6 - 4 = 2\)
\(2 = |(3i + 4j)||(2i - j)| \cos \theta\)
\(|3i + 4j| = \sqrt{3^{2} + 4^{2}} = \sqrt{25} = 5\)
\(|2i - j| = \sqrt{2^{2} + (-1)^{2}} = \sqrt{5}\)
\(2 = 5(\sqrt{5})(\cos \theta)\)
\(\cos \theta = \frac{2}{5\sqrt{5}} = 0.08\sqrt{5}\)
\(\theta = \cos^{-1} 0.1788 = 79.7°\)
Given that \(\log_{2} y^{\frac{1}{2}} = \log_{5} 125\), find the value of y
16
25
36
64
Correct answer is D
\(\log_{2} y^{\frac{1}{2}} = \log_{5} 125\)
\(\log_{2} y^{\frac{1}{2}} = \log_{5} 5^{3} = 3\log_{5} 5 = 3\)
\(\log_{2} y^{\frac{1}{2}} = 3 \implies y^{\frac{1}{2}} = 2^{3} = 8\)
\(y = 8^{2} = 64\)
(1, 2)
(1, 1)
(1, -1)
(1, -2)
Correct answer is C
\(y = 4x^{3} + kx^{2} - 6x + 4\)
\(\frac{\mathrm d y}{\mathrm d x} = 12x^{2} + 2kx - 6\)
At P(1, m)
\(\frac{\mathrm d y}{\mathrm d x} = 12 + 2k - 6 = 0\) (parallel to the x- axis)
\(6 + 2k = 0 \implies k = -3\)
\(P(1, m) \implies m = 4(1^{3}) - 3(1^{2}) - 6(1) + 4)
= -1
P = (1, -1)